arXiv subjects

Itzhak Tamo

Publications and source records attributed to Itzhak Tamo.

At least 37 records · Page 2Linked to original sources

arXiv2024

Repairing Reed-Solomon Codes over Prime Fields via Exponential Sums

This paper presents two repair schemes for low-rate Reed-Solomon (RS) codes over prime fields that can repair any node by downloading a constant number of bits from each surviving node. The total bandwidth resulting from these schemes is greater than that incurred during trivial repair; however, this is particularly relevant in the context of leakage-resilient secret sharing. In that framework, our results provide attacks showing that $k$-out-of-$n$ Shamir's Secret Sharing over prime fields for small $k$ is not leakage-resilient, even when the parties leak only a constant number of bits. To the best of our knowledge, these are the first such attacks. Our results are derived from a novel connection between exponential sums and the repair of RS codes. Specifically, we establish that non-trivial bounds on certain exponential sums imply the existence of explicit nonlinear repair schemes for RS codes over prime fields.

cs.IT↗

arXiv2024

Explicit Subcodes of Reed-Solomon Codes that Efficiently Achieve List Decoding Capacity

In this paper, we introduce a novel explicit family of subcodes of Reed-Solomon (RS) codes that efficiently achieve list decoding capacity with a constant output list size. Our approach builds upon the idea of large linear subcodes of RS codes evaluated on a subfield, similar to the method employed by Guruswami and Xing (STOC 2013). However, our approach diverges by leveraging the idea of {\it permuted product codes}, thereby simplifying the construction by avoiding the need of {\it subspace designs}. Specifically, the codes are constructed by initially forming the tensor product of two RS codes with carefully selected evaluation sets, followed by specific cyclic shifts to the codeword rows. This process results in each codeword column being treated as an individual coordinate, reminiscent of prior capacity-achieving codes, such as folded RS codes and univariate multiplicity codes. This construction is easily shown to be a subcode of an interleaved RS code, equivalently, an RS code evaluated on a subfield. Alternatively, the codes can be constructed by the evaluation of bivariate polynomials over orbits generated by \emph{two} affine transformations with coprime orders, extending the earlier use of a single affine transformation in folded RS codes and the recent affine folded RS codes introduced by Bhandari {\it et al.} (IEEE T-IT, Feb.~2024). While our codes require large, yet constant characteristic, the two affine transformations facilitate achieving code length equal to the field size, without the restriction of the field being prime, contrasting with univariate multiplicity codes.

cs.IT↗

arXiv2023

Tighter List-Size Bounds for List-Decoding and Recovery of Folded Reed-Solomon and Multiplicity Codes

Folded Reed-Solomon (FRS) and univariate multiplicity codes are prominent polynomial codes over finite fields, renowned for achieving list decoding capacity. These codes have found a wide range of applications beyond the traditional scope of coding theory. In this paper, we introduce improved bounds on the list size for list decoding of these codes, achieved through a more streamlined proof method. Additionally, we refine an existing randomized algorithm to output the codewords on the list, enhancing its success probability and reducing its running time. Lastly, we establish list-size bounds for a fixed decoding parameter. Notably, our results demonstrate that FRS codes asymptotically attain the generalized Singleton bound for a list of size $2$ over a relatively small alphabet, marking the first explicit instance of a code with this property.

cs.IT↗

arXiv2023

Improved List-Decodability of Reed--Solomon Codes via Tree Packings

This paper shows that there exist Reed--Solomon (RS) codes, over \black{exponentially} large finite fields \black{in the code length}, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving the list-decoding capacity. In particular, we show that for any $ε\in (0,1]$ there exist RS codes with rate $Ω(\fracε{\log(1/ε)+1})$ that are list-decodable from radius of $1-ε$. We generalize this result to list-recovery, showing that there exist $(1 - ε, \ell, O(\ell/ε))$-list-recoverable RS codes with rate $Ω\left( \fracε{\sqrt{\ell} (\log(1/ε)+1)} \right)$. Along the way we use our techniques to give a new proof of a result of Blackburn on optimal linear perfect hash matrices, and strengthen it to obtain a construction of strongly perfect hash matrices. To derive the results in this paper we show a surprising connection of the above problems to graph theory, and in particular to the tree packing theorem of Nash-Williams and Tutte. We also state a new conjecture that generalizes the tree-packing theorem to hypergraphs, and show that if this conjecture holds, then there would exist RS codes that are \em optimally \em (non-asymptotically) list-decodable.

cs.IT↗

arXiv2023

Points-Polynomials Incidence Theorem with Applications to Coding Theory

This paper focuses on incidences over finite fields, extending to higher degrees a result by Vinh \cite{VINH20111177} on the number of point-line incidences in the plane $\mathbb{F}^2$, where $\mathbb{F}$ is a finite field. Specifically, we present a bound on the number of incidences between points and polynomials of bounded degree in $\mathbb{F}^2$. Our approach employs a singular value decomposition of the incidence matrix between points and polynomials, coupled with an analysis of the related group algebras. This bound is then applied to coding theory, specifically to the problem of average-radius list decoding of Reed-Solomon (RS) codes. We demonstrate that RS codes of certain lengths are average-radius list-decodable with a constant list size, which is dependent on the code rate and the distance from the Johnson radius. While a constant list size for list-decoding of RS codes in this regime was previously established, its existence for the stronger notion of average-radius list-decoding was not known to exist.

cs.IT↗

arXiv2023

Access-Redundancy Tradeoffs in Quantized Linear Computations

Linear real-valued computations over distributed datasets are common in many applications, most notably as part of machine learning inference. In particular, linear computations that are quantized, i.e., where the coefficients are restricted to a predetermined set of values (such as $\pm 1$), have gained increasing interest lately due to their role in efficient, robust, or private machine learning models. Given a dataset to store in a distributed system, we wish to encode it so that all such computations could be conducted by accessing a small number of servers, called the access parameter of the system. Doing so relieves the remaining servers to execute other tasks. Minimizing the access parameter gives rise to an access-redundancy tradeoff, where a smaller access parameter requires more redundancy in the system, and vice versa. In this paper, we study this tradeoff and provide several explicit low-access schemes for $\{\pm1\}$ quantized linear computations based on covering codes in a novel way. While the connection to covering codes has been observed in the past, our results strictly outperform the state-of-the-art for two-valued linear computations. We further show that the same storage scheme can be used to retrieve any linear combination with two distinct coefficients -- regardless of what those coefficients are -- with the same access parameter. This universality result is then extended to all possible quantizations with any number of values; while the storage remains identical, the access parameter increases according to a new additive-combinatorics property we call coefficient complexity. We then turn to study the coefficient complexity -- we characterize the complexity of small sets of coefficients, provide bounds, and identify coefficient sets having the highest and lowest complexity.

cs.IT↗

arXiv2022

Multivariate Public Key Cryptosystem from Sidon Spaces

A Sidon space is a subspace of an extension field over a base field in which the product of any two elements can be factored uniquely, up to constants. This paper proposes a new public-key cryptosystem of the multivariate type which is based on Sidon spaces, and has the potential to remain secure even if quantum supremacy is attained. This system, whose security relies on the hardness of the well-known MinRank problem, is shown to be resilient to several straightforward algebraic attacks. In particular, it is proved that the two popular attacks on the MinRank problem, the kernel attack, and the minor attack, succeed only with exponentially small probability. The system is implemented in software, and its hardness is demonstrated experimentally.

cs.CR↗

arXiv2022

Nonlinear Repair of Reed-Solomon Codes

The problem of repairing linear codes and, in particular, Reed Solomon (RS) codes has attracted a lot of attention in recent years due to their extreme importance to distributed storage systems. In this problem, a failed code symbol (node) needs to be repaired by downloading as little information as possible from a subset of the remaining nodes. By now, there are examples of RS codes that have efficient repair schemes, and some even attain the cut-set bound. However, these schemes fall short in several aspects; they require a considerable field extension degree. They do not provide any nontrivial repair scheme over prime fields. Lastly, they are all linear repairs, i.e., the computed functions are linear over the base field. Motivated by these and by a question raised in [GW17] on the power of nonlinear repair schemes, we study the problem of nonlinear repair schemes of RS codes. Our main results are the first nonlinear repair scheme of RS codes with asymptotically optimal repair bandwidth (asymptotically matching the cut-set bound). This is the first example of a nonlinear repair scheme of any code and also the first example that a nonlinear repair scheme can outperform all linear ones. Lastly, we show that the cut-set bound for RS codes is not tight over prime fields by proving a tighter bound, using additive combinatorics ideas.

cs.IT↗

arXiv2022

Reed Solomon Codes Against Adversarial Insertions and Deletions

In this work, we study the performance of Reed--Solomon codes against adversarial insertion-deletion (insdel) errors. We prove that over fields of size $n^{O(k)}$ there are $[n,k]$ Reed-Solomon codes that can decode from $n-2k+1$ insdel errors and hence attain the half-Singleton bound. We also give a deterministic construction of such codes over much larger fields (of size $n^{k^{O(k)}}$). Nevertheless, for $k=O(\log n /\log\log n)$ our construction runs in polynomial time. For the special case $k=2$, which received a lot of attention in the literature, we construct an $[n,2]$ Reed-Solomon code over a field of size $O(n^4)$ that can decode from $n-3$ insdel errors. Earlier constructions required an exponential field size. Lastly, we prove that any such construction requires a field of size $Ω(n^3)$.

cs.IT↗

arXiv2022

Explicit and Efficient Constructions of linear Codes Against Adversarial Insertions and Deletions

In this work, we study linear error-correcting codes against adversarial insertion-deletion (insdel) errors, a topic that has recently gained a lot of attention. We construct linear codes over $\mathbb{F}_q$, for $q=\text{poly}(1/\varepsilon)$, that can efficiently decode from a $δ$ fraction of insdel errors and have rate $(1-4δ)/8-\varepsilon$. We also show that by allowing codes over $\mathbb{F}_{q^2}$ that are linear over $\mathbb{F}_q$, we can improve the rate to $(1-δ)/4-\varepsilon$ while not sacrificing efficiency. Using this latter result, we construct fully linear codes over $\mathbb{F}_2$ that can efficiently correct up to $δ< 1/54$ fraction of deletions and have rate $R = (1-54\cdot δ)/1216$. Cheng, Guruswami, Haeupler, and Li [CGHL21] constructed codes with (extremely small) rates bounded away from zero that can correct up to a $δ< 1/400$ fraction of insdel errors. They also posed the problem of constructing linear codes that get close to the half-Singleton bound (proved in [CGHL21]) over small fields. Thus, our results significantly improve their construction and get much closer to the bound.

cs.IT↗

arXiv2021

Singleton-type bounds for list-decoding and list-recovery, and related results

List-decoding and list-recovery are important generalizations of unique decoding that received considerable attention over the years. However, the optimal trade-off among list-decoding (resp. list-recovery) radius, list size, and the code rate are not fully understood in both problems. This paper takes a step towards this direction when the list size is a given constant and the alphabet size is large (as a function of the code length). We prove a new Singleton-type upper bound for list-decodable codes, which improves upon the previously known bound by roughly a factor of $1/L$, where $L$ is the list size. We also prove a Singleton-type upper bound for list-recoverable codes, which is to the best of our knowledge, the first such bound for list-recovery. We apply these results to obtain new lower bounds that are optimal up to a multiplicative constant on the list size for list-decodable and list-recoverable codes with rates approaching capacity. Moreover, we show that list-decodable \emph{nonlinear} codes can strictly outperform list-decodable linear codes. More precisely, we show that there is a gap for a wide range of parameters, which grows fast with the alphabet size, between the size of the largest list-decodable nonlinear code and the size of the largest list-decodable linear codes. This is achieved by a novel connection between list-decoding and the notion of sparse hypergraphs in extremal combinatorics. We remark that such a gap is not known to exist in the problem of unique decoding. Lastly, we show that list-decodability or recoverability of codes implies in some sense good unique decodability.

cs.IT↗

arXiv2021

A construction of maximally recoverable codes

We construct a family of linear maximally recoverable codes with locality $r$ and dimension $r+1.$ For codes of length $n$ with $r\approx n^α, 0\leα\le 1$ the code alphabet is of the order $n^{1+3α},$ which improves upon the previously known constructions of maximally recoverable codes.

cs.IT↗

arXiv2021

List-decoding and list-recovery of Reed-Solomon codes beyond the Johnson radius for any rate

Understanding the limits of list-decoding and list-recovery of Reed-Solomon (RS) codes is of prime interest in coding theory and has attracted a lot of attention in recent decades. However, the best possible parameters for these problems are still unknown, and in this paper, we take a step in this direction. We show the existence of RS codes that are list-decodable or list-recoverable beyond the Johnson radius for \emph{any} rate, with a polynomial field size in the block length. In particular, we show that for any $ε\in (0,1)$ there exist RS codes that are list-decodable from radius $1-ε$ and rate less than $\fracε{2-ε}$, with constant list size. We deduce our results by extending and strengthening a recent result of Ferber, Kwan, and Sauermann on puncturing codes with large minimum distance and by utilizing the underlying code's linearity.

cs.IT↗

arXiv2021

Bounds on the Capacity of PIR over Graphs

In the private information retrieval (PIR) problem, a user wants to retrieve a file from a database without revealing any information about the desired file's identity to the servers that store the database. In this paper, we study the PIR capacity of a graph-based replication system, in which each file is stored on two distinct servers according to an underlying graph. This paper aims to provide upper and lower bounds to the PIR capacity of graphs via various graph properties. In particular, we provide several upper bounds on the PIR capacity that apply to all graphs. We further improve the bounds for specific graph families (which turn out to be tight in certain cases) by utilizing the underlying graph structure. For the lower bounds, we establish optimal rate PIR retrieval schemes for star graphs via edge-coloring techniques. Lastly, we provide an improved PIR scheme for complete graphs, which implies an improved general lower bound on all graphs' PIR capacity.

cs.IT↗

arXiv2020

Repairing Reed-Solomon Codes Evaluated on Subspaces

We consider the repair problem for Reed--Solomon (RS) codes, evaluated on an $\mathbb{F}_q$-linear subspace $U\subseteq\mathbb{F}_{q^m}$ of dimension $d$, where $q$ is a prime power, $m$ is a positive integer, and $\mathbb{F}_q$ is the Galois field of size $q$. For the case of $q\geq 3$, we show the existence of a linear repair scheme for the RS code of length $n=q^d$ and codimension $q^s$, $s< d$, evaluated on $U$, in which each of the $n-1$ surviving nodes transmits only $r$ symbols of $\mathbb{F}_q$, provided that $ms\geq d(m-r)$. For the case of $q=2$, we prove a similar result, with some restrictions on the evaluation linear subspace $U$. Our proof is based on a probabilistic argument, however the result is not merely an existence result; the success probability is fairly large (at least $1/3$) and there is a simple criterion for checking the validity of the randomly chosen linear repair scheme. Our result extend the construction of Dau--Milenkovich to the range $r<m-s$, for a wide range of parameters.

cs.IT↗

arXiv2020

Minimum Guesswork with an Unreliable Oracle

We study a guessing game where Alice holds a discrete random variable $X$, and Bob tries to sequentially guess its value. Before the game begins, Bob can obtain side-information about $X$ by asking an oracle, Carole, any binary question of his choosing. Carole's answer is however unreliable, and is incorrect with probability $ε$. We show that Bob should always ask Carole whether the index of $X$ is odd or even with respect to a descending order of probabilities -- this question simultaneously minimizes all the guessing moments for any value of $ε$. In particular, this result settles a conjecture of Burin and Shayevitz. We further consider a more general setup where Bob can ask a multiple-choice $M$-ary question, and then observe Carole's answer through a noisy channel. When the channel is completely symmetric, i.e., when Carole decides whether to lie regardless of Bob's question and has no preference when she lies, a similar question about the ordered index of $X$ (modulo $M$) is optimal. Interestingly however, the problem of testing whether a given question is optimal appears to be generally difficult in other symmetric channels. We provide supporting evidence for this difficulty, by showing that a core property required in our proofs becomes NP-hard to test in the general $M$-ary case. We establish this hardness result via a reduction from the problem of testing whether a system of modular difference disequations has a solution, which we prove to be NP-hard for $M\geq 3$.

cs.DM↗

arXiv2020

New Turán exponents for two extremal hypergraph problems

An $r$-uniform hypergraph is called $t$-cancellative if for any $t+2$ distinct edges $A_1,\ldots,A_t,B,C$, it holds that $(\cup_{i=1}^t A_i)\cup B\neq (\cup_{i=1}^t A_i)\cup C$. It is called $t$-union-free if for any two distinct subsets $\mathcal{A},\mathcal{B}$, each consisting of at most $t$ edges, it holds that $\cup_{A\in\mathcal{A}} A\neq \cup_{B\in\mathcal{B}} B$. Let $C_t(n,r)$ (resp. $U_t(n,r)$) denote the maximum number of edges of a $t$-cancellative (resp. $t$-union-free) $r$-uniform hypergraph on $n$ vertices. Among other results, we show that for fixed $r\ge 3,t\ge 3$ and $n\rightarrow\infty$ $$Ω(n^{\lfloor\frac{2r}{t+2}\rfloor+\frac{2r\pmod{t+2}}{t+1}})=C_t(n,r)=O(n^{\lceil\frac{r}{\lfloor t/2\rfloor+1}\rceil})\text{ and } Ω(n^{\frac{r}{t-1}})=U_t(n,r)=O(n^{\lceil\frac{r}{t-1}\rceil}),$$ thereby significantly narrowing the gap between the previously known lower and upper bounds. In particular, we determine the Turán exponent of $C_t(n,r)$ when $2\mid t \text{ and } (t/2+1)\mid r$, and of $U_t(n,r)$ when $(t-1)\mid r$. The main tool used in proving the two lower bounds is a novel connection between these problems and sparse hypergraphs.

math.CO↗

arXiv2020

Sparse Hypergraphs with Applications to Coding Theory

For fixed integers $r\ge 3,e\ge 3,v\ge r+1$, an $r$-uniform hypergraph is called $\mathscr{G}_r(v,e)$-free if the union of any $e$ distinct edges contains at least $v+1$ vertices. Brown, Erdős and Sós showed that the maximum number of edges of such a hypergraph on $n$ vertices, denoted as $f_r(n,v,e)$, satisfies $$Ω(n^{\frac{er-v}{e-1}})=f_r(n,v,e)=\mathcal{O}(n^{\lceil\frac{er-v}{e-1}\rceil}).$$ For $e-1\mid er-v$, the lower bound matches the upper bound up to a constant factor; whereas for $e-1\nmid er-v$, in general it is a notoriously hard problem to determine the correct exponent of $n$. Among other results, we improve the above lower bound by showing that $$f_r(n,v,e)=Ω(n^{\frac{er-v}{e-1}}(\log n)^{\frac{1}{e-1}})$$ for any $r,e,v$ satisfying $\gcd(e-1,er-v)=1$. The hypergraph we constructed is in fact $\mathscr{G}_r(ir-\lceil\frac{(i-1)(er-v)}{e-1}\rceil,i)$-free for every $2\le i\le e$, and it has several interesting applications in Coding Theory. The proof of the new lower bound is based on a novel application of the lower bound on the hypergraph independence number due to Duke, Lefmann, and R{ö}dl.

math.CO↗